Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s − 4·7-s + 9-s − 11-s + 4·13-s + 2·15-s − 8·21-s − 6·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s − 2·33-s − 4·35-s − 2·37-s + 8·39-s − 6·41-s + 8·43-s + 45-s + 6·47-s + 9·49-s + 6·53-s − 55-s + 12·59-s + 2·61-s − 4·63-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s − 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s + 1.56·59-s + 0.256·61-s − 0.503·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(79420\)    =    \(2^{2} \cdot 5 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{79420} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 79420,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.536243410\)
\(L(\frac12)\)  \(\approx\)  \(2.536243410\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.87488066394318, −13.60043451993836, −13.16987545115198, −12.61111005632788, −12.28864679647435, −11.48806130611251, −10.91135579013673, −10.28387642733999, −9.912830563758186, −9.462725967914811, −8.943168512738113, −8.425065363430565, −8.194132778313410, −7.209094848315181, −6.969319928852918, −6.216602829717158, −5.762056623239748, −5.359817608675505, −4.162181880596534, −3.825968726624951, −3.324195508779203, −2.667623518117892, −2.270190858041022, −1.446082270756031, −0.4653615643970558, 0.4653615643970558, 1.446082270756031, 2.270190858041022, 2.667623518117892, 3.324195508779203, 3.825968726624951, 4.162181880596534, 5.359817608675505, 5.762056623239748, 6.216602829717158, 6.969319928852918, 7.209094848315181, 8.194132778313410, 8.425065363430565, 8.943168512738113, 9.462725967914811, 9.912830563758186, 10.28387642733999, 10.91135579013673, 11.48806130611251, 12.28864679647435, 12.61111005632788, 13.16987545115198, 13.60043451993836, 13.87488066394318

Graph of the $Z$-function along the critical line